Detailed examination of computed particle trajectories has revealed a complexity and disorder that is of increasing interest to accelerator specialists. To introduce this topic, the author would like you to consider for a moment the analysis of synchrotron oscillations for a particle in a coasting beam, regarded as a problem in one degree of freedom. A simple analysis replaces the electric field of the RF-v cavity system by a traveling wave, having the speed of a synchronous reference particle, and leads to a pair of differential equations of the form dy/dn = -K sin {pi}x, (1A) where y measures the fractional departure of energy from the reference value {pi}x measures the electrical phase angle at which the particle traverses the cavity, and K is proportional to the cavity voltage; and dx/dn = {lambda}{prime}y, (1b) in which {lambda}{prime} is proportional to the change of revolution period with respect to particle energy. It will be recognized that these equations can be derived from a Hamiltonian function H = (1/2){lambda}{prime}y{sup 2}-(K/{pi})cos {pi}x. (2) Because this Hamiltonian function does not contain the independent variable explicitly, it will constitute a constant of the motion and possible trajectories in the x,y phase space will be just the curves defined by H = Constant, namely the familiar simple curves in phase space that are characteristic of a physical (non-linear) pendulum
